feat(web): list background tasks in the session header

The task registry has run every background bash, pwsh, pty-send, and
one-shot subagent since it landed, but only the model could read it: a
human at the Web client could not see that a build was running, tell a
finished task from a stuck one, or find its outcome anywhere but the
`run_in_background` tool card that printed an id and never updated.

Task state now reaches the browser as one whole-snapshot `session/tasks`
mux frame per session, pushed at every registry commit that changes what
that session can see. `TaskService` gains `onTasksChanged`, which is
owner-granular because owner-disposal removal is a change no per-task
record can express. The carrier reads the exact owner the listener hands
it, so a push stays correct while that scope tears down, and reads the
baseline through the non-resuming `ctx.agents.get` so listing never
revives a cold session. The client keeps a last-wins mirror on
`SessionListState`, and a new `dsh-client-ui-task` package renders it
beside the subagent catalog — rendering nothing at all until the session
has a task, so an ordinary conversation grows no new chrome.

Streamed per-task output and human-initiated cancellation are separate
phases; the note records why neither has to undo this channel, and why
no Web path may call the consuming `ctx.tasks.read()`.

packages/client/ui-conversation/tests/chat-view.spec.tsx

@@ -110,7 +110,7 @@ const compaction = (over: Partial<CompactionSummaryNode> = {}): CompactionSummar
 /** Empty sessions-list hook for the global standard-kit seat. */
 function emptySessions() {
   const store = createSnapshotStore<SessionListState>(
-    { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, currentAddress: undefined })
+    { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined })
   return bindSnapshotSelector(store)
 }
 

packages/client/ui-conversation/tests/enter-behavior-row.spec.tsx

@@ -16,7 +16,7 @@ afterEach(() => {
 
 function emptySessions() {
   return bindSnapshotSelector(createSnapshotStore<SessionListState>({
-    ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, currentAddress: undefined,
+    ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined,
   }))
 }
 

packages/client/ui-conversation/tests/gate-branch-tails.spec.tsx

@@ -100,7 +100,7 @@ describe('render branch tails', () => {
     const chat = createChatStore().create()
     chat.actions.select({ turnSeq: 1, callId: 'ghost' } satisfies SelectionTarget)
     const emptyList = createSnapshotStore<SessionListState>(
-      { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, currentAddress: undefined })
+      { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined })
     const emptyWorkspaces = createSnapshotStore<WorkspaceListState>({
       items: [], archivedSessionIds: [], state: 'idle', phase: 'ready', error: null,
       baselinesReady: true, recentWorkspaceId: undefined,
@@ -149,7 +149,7 @@ describe('render branch tails', () => {
     const chat = createChatStore().create()
     chat.actions.select({ turnSeq: 9, callId: 'p1:code:1:code:1', toolName: 'read' } satisfies SelectionTarget)
     const emptyList = createSnapshotStore<SessionListState>(
-      { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, currentAddress: undefined })
+      { ids: [], byId: {}, current: undefined, phase: 'ready', subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined })
     const emptyWorkspaces = createSnapshotStore<WorkspaceListState>({
       items: [], archivedSessionIds: [], state: 'idle', phase: 'ready', error: null,
       baselinesReady: true, recentWorkspaceId: undefined,

packages/client/ui-conversation/tests/input-bar.spec.tsx

@@ -109,7 +109,7 @@ function bench(over?: BenchOptions) {
     useSession: bindSnapshotSelector(session),
     useSessions: bindSnapshotSelector(createSnapshotStore({
       ids: [], byId: {}, current: undefined, phase: 'ready',
-      subagentsByParent: {}, currentAddress: undefined,
+      subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined,
     })),
     useWorkspaces: bindSnapshotSelector(createSnapshotStore({
       items: [], archivedSessionIds: [], state: 'idle', phase: 'ready', error: null,

packages/client/ui-conversation/tests/input-matrix.spec.tsx

@@ -37,7 +37,7 @@ function mountBar(shell: SessionInputShell, over?: { running?: boolean; disabled
     useSession: bindSnapshotSelector(session),
     useSessions: bindSnapshotSelector(createSnapshotStore({
       ids: [], byId: {}, current: undefined, phase: 'ready',
-      subagentsByParent: {}, currentAddress: undefined,
+      subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined,
     })),
     useWorkspaces: bindSnapshotSelector(createSnapshotStore({
       items: [], archivedSessionIds: [], state: 'idle', phase: 'ready', error: null,

packages/client/ui-conversation/tests/input-scenarios.spec.tsx

@@ -123,7 +123,7 @@ async function scopedBench(register?: (slash: SlashService) => void) {
     useSession: bindSnapshotSelector(sessionStore),
     useSessions: bindSnapshotSelector(createSnapshotStore({
       ids: [], byId: {}, current: undefined, phase: 'ready',
-      subagentsByParent: {}, currentAddress: undefined,
+      subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined,
     })),
     useWorkspaces: bindSnapshotSelector(createSnapshotStore({
       items: [], archivedSessionIds: [], state: 'idle', phase: 'ready', error: null,

packages/client/ui-conversation/tests/skeleton.spec.tsx

@@ -107,7 +107,7 @@ function mount(
     ids: listed ? [root, SID] : [root],
     byId: { [root]: rootRow, ...listed && { [SID]: childRow } },
     current: SID,
-    phase: 'ready', subagentsByParent: {}, currentAddress: undefined,
+    phase: 'ready', subagentsByParent: {}, tasksBySession: {}, currentAddress: undefined,
   })
   const workspaces = createSnapshotStore<WorkspaceListState>(workspaceState(workspaceRows))
   const session = createSnapshotStore<ConversationSnapshot>(snapshot)